The original question was: If I’m moving at the speed of light towards you and I throw a tennis ball at you (at, say, 5 m/s), what do you observe? Will I hit you first, the tennis ball, or both at the same time, or will something else happen entirely? The problem that I’m having thinking about this is that if the tennis ball did hit you first, then it would have been moving faster than the speed of light (relative to you). On the other hand, if it were moving at the speed of light relative to you, then it was moving at the same speed as me relative to you, thus both myself and the tennis ball will hit you simultaneously. If this is the case, however, then the tennis ball would have been moving at a speed of 0 m/s with respect to me.
Mathematician: First of all, let me point out that you will never travel at the speed of light (see this for details). It would take an infinite amount of energy to get anything with mass (e.g. you or your mama) going at that speed. Burning all the oil (and plants, and animals) on our planet and converting them into kinetic energy would get you going really fast, but would give you exactly 0% of the total energy that you would actually need to get going that speed (since any number divided by infinity is zero). But that doesn’t mean that it wouldn’t be fun to speculate about what would happen if you were going at light speed.
Due to a relativistic effect known as time dilation, the faster that you move with respect to some object O, the more time slows down for the object O (from your perspective). This isn’t just an issue of you seeing clocks attached to O tick slowly, they actually DO tick slowly from your vantage point (no experiment you could possibly do would conclude otherwise). There is symmetry though. You moving past O at 10,000 miles per hour is indistinguishable (as far as the laws of physics are concerned) from O moving past you at 10,000 miles per hour. That means, from the perspective of a person strapped to O, clocks attached to you are ticking slowly (i.e. your time is slowed down). As you approach the speed of light (with respect to O), this time dilation effect becomes more and more pronounced. When you are exactly AT the speed of light (impossible, but bare with me) no time whatsoever will elapse for O (i.e. a clock strapped to O will stop ticking completely) from your perspective. The upshot of this is that you’ll get wherever you are going without witnessing any time pass for anything not moving along with you. One reason this is really trippy is because if we view light coming from a distance object (such as a far away sun), from our perspective it might have taken years to get from us. But from the perspective of the photon (i.e. the light particle) no time will have elapsed on the journey! Yes, true physics is even weirder than crazy person made up physics.
A possibly even wackier effect crops up as the result of length contraction (another consequence of relativity). If you move towards object O at a fast speed you will notice that O will be compressed (i.e. shrunk) along the direction of your motion. So if O is a hippo, and you are going fast enough, it will look like one damn flat hippo.
As you approach the speed of light, this effect becomes increasingly pronounced, and at at the speed of light itself O will have zero length in the direction you are traveling. In particular, if you are on a straight race track, and traveling at the speed of light, the race track will be compressed to zero length so that the starting line and the finishing line will be on top of each other. The race will be over as soon as it begins.
Another consequence of light speed travel is that you’d become the most dangerous thing imaginable (move over, Chuck). Since your mass is positive, infinite speed implies that your momentum is infinite. Hence, if you crashed into anything (and you would…after all, from your perspective the universe is flatter than a pancake in the direction you’re heading) it would get hurtled at insane speeds (since it would absorb some of your momentum). Of course, you’d also be dead pretty much instantly as you collided with object after object (each traveling at the speed of light with respect to you). And no, armor wouldn’t help.
Okay, so now to address the original question. What would happen if while traveling at the speed of light towards me you attempted to throw a ball at me? The answer is that you would have no time to actually do the throwing, because from your perspective you would run into me instantly. At which point, if I had any ninja skills, I would probably break those out. If you were traveling at near the speed of light (with respect to me), but not quite at it, and then threw the ball at 5 m/s (with respect to you) in my direction, the velocities would not simply add like you would expect based on Newtonian mechanics. Instead, you’d have to apply relativistic velocity addition which is a bit more complicated. In particular, the speed of the ball with respect to me will be less than the sum of your speed and 5 m/s. At low speeds this effect is not noticeable (speeds are additive to very close approximation), but at speeds close to the speed of light the effect becomes very pronounced.